Similar to the 2D quantum Hall effect on a lattice, the topological pump is characterized by a 2D Chern number and exhibits quantized charge transport for both rational and irrational \(\tau \) . We, now, focus instead on the direct topological characterization of the quasiperiodic chain using Chern numbers, i.e., without the need to use the parameter \(\phi \) , where we summarize and repeat the proof shown in Ref. [1]}, cf. also Refs. [2]}, [3]}. To do so, we employ the approximant path described above, and place at each iteration the chain on a ring (periodic boundary conditions) of length \(q_n\) . We also introduce a flux \(\theta \)  (phase twist) threading the ring, cf. Fig. REF . The periodic model has the same spectral gaps as those of the original chain up to small corrections [4]}, whereas in the thermodynamic limit the effect of \(\theta \) 
completely vanishes [4]}, [6]}.
